We now characterize an optimal relaxed policy \(\pi ^\star _{\mathrm {R}}\)  for (P2). If
\(\frac{1}{K}\sum _{k =1}^{K}\bar{J}_{\pi _{\mathrm {R},\mu ^*,k}^\star } = \Gamma \) , then, \(\pi _{\mathrm {R},\mu ^*,k}^\star \) , \({k\in \mathcal {K}}\) , form an optimal policy for (P2), i.e., \(\pi _{\mathrm {R}}^\star = \pi _{\mathrm {R},\mu ^*}^\star \) .
Otherwise, \(\pi _{\mathrm {R}}^\star \)  is a mixture of two deterministic policies \(\pi _{\mathrm {R},\mu ^{*-}}^\star \)  and \(\pi _{\mathrm {R},\mu ^{*+}}^\star \) , which are defined by [1]}
\(\pi _{\mathrm {R},\mu ^{*-}}^\star \triangleq \lim _{\mu \rightarrow \mu ^{*-}} \pi _{\mathrm {R},\mu }^\star ~\mathrm {and}~\pi _{\mathrm {R},\mu ^{*+}}^\star \triangleq \lim _{\mu \rightarrow \mu ^{*+}} \pi _{\mathrm {R},\mu }^\star ,\) 
